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1

Photoelectron ionization spectra in a system interacting with a neighbor atom

Jan Perina Jr.,1 Antonın Luks,2 Wieslaw Leonski,3 and Vlasta Perinova2

1Institute of Physics of AS CR, Joint Laboratory of Optics,

17. listopadu 50a, 772 07 Olomouc, Czech Republic2Palacky University, RCPTM, Joint Laboratory of Optics,

17. listopadu 12, 771 46 Olomouc, Czech Republic3Quantum Optics and Engineering Division, Institute of Physics,

University of Zielona Gora, Prof. Z. Szafrana 4a, 65-516 Zielona Gora, Poland

Photoelectron ionization spectra of a system interacting with a neighbor two-level atom are in-vestigated using the Laplace-transform method. These spectra are typically composed of severalpeaks. Photoelectron ionization spectra conditioned by the measurement on the two-level atomshow oscillations at the Rabi frequency. The presence of spectral zeros occurring periodically withthe Rabi period is predicted. This phenomenon is analyzed in detail.

PACS numbers: 32.80.-t,33.80.Eh,34.20.-b

I. INTRODUCTION

Ionization of an atom represents one of the most im-portant physical effects. In this process an electron in abound state at an atom is moved into a free state. Thereexist several mechanisms leading to ionization, e.g., ther-mal ionization or ionization in a strong static electric ormagnetic field. Optical ionization is probably the mostimportant, because it allows a versatile diagnostics of theelectron system of the studied atom. At the quantumlevel, an electron in the bound state moves to a free stateafter absorbing one photon (or several photons) from theoptical field. This implies that photoelectron ionizationspectra depend strongly on the frequency of the opticalfield.

Shapes of the photoelectron ionization spectra are in-fluenced by many factors. Among them, the presence ofdiscrete bound excited electron states at the atom playsa dominant role as was pointed out for the first time byFano [1]. The presence of such excited bound states thusforms additional ionization paths that compete with di-rect ionization. Along these additional paths, an electronmoves from its ground bound state to an excited boundstate due to the optical field first and, subsequently, theCoulomb configurational interaction transfers the elec-tron into a free state. Interference among direct andindirect ionization paths determines the shapes of pho-toelectron ionization spectra [1–3]. Fano has shown thatthere exist free states that cannot be populated owingto a completely destructive interference of the ionizationpaths. This effect is referred to as the presence of Fano ze-ros that do not depend on the optical-field strength. Ex-perimental evidence of the Fano zeros has been brought,e.g., in [4]. In general, there exist N Fano zeros in asystem with N discrete levels [1]. Special attention hasbeen devoted to auto-ionization systems with two levelsin various configurations [5–8]. The interaction of auto-ionization systems with quantum optical fields has beenaddressed in [9]. Also the transparency for ultra-shortoptical pulses has been predicted in auto-ionization sys-

tems [10]. Auto-ionization systems can even slow-downthe propagating light under certain conditions [11]. Sim-ilarly as other physical effects, the dynamics of ionizationcan be tailored using the Zeno and anti-Zeno effects [12].Moreover, quite recently Chu and Lin [13] have proposedthe method of ultra-fast Fano resonances in the descrip-tion of dynamics using the model involving ultra-shortlaser pulses. It should be stressed out that the problemof Fano resonances is not restricted to the atomic physicsonly. There are numerous papers in the field of solidstate physics, especially devoted to the systems contain-ing quantum dots and nano-particles. For instance, theinfluence of the effect of Fano resonances on the photonstatistics has been discussed in [14]. A wider overview ofliterature devoted to the Fano resonances in nano-physicscan be found in [15].

Here, we study the role of bound excited states locatedat a neighbor atom in the formation of photoelectron ion-ization spectra [16]. An electron in a bound state of theneighbor atom interacts with an electron at the ionizationatom through the dipole-dipole interaction that leads toenergy transfer between two electrons [17]. This typeof interaction (energy transfer) qualitatively differs fromthat embedded in the Fano model (electron transfer). Asa consequence, the Fano zeros do not exist in this model.However, another type of spectral zeros has been discov-ered here and is named dynamical zeros. Such dynamicalzeros occur periodically with the Rabi frequency in theconditional photoelectron ionization spectra. If boundauto-ionizing states in the ionization system are present,both the Fano and dynamical zeros can be found in pho-toelectron ionization spectra, as will be discussed in aconsecutive publication. Suitable candidates for an ex-perimental verification of the presented theory are molec-ular condensates, in which the electrons localized at dif-ferent molecules interact through the dipole-dipole inter-action [17]. The developed theory can also be testified us-ing quantum dots and other semiconductor heterostruc-tures [15].

The paper is organized as follows. In Sec. II, the modelHamiltonian is given and the corresponding dynamical

2

µaαL

µαL

J

|0〉a |0〉b

|1〉a

|E)

Ea

E

FIG. 1. Scheme of the ionization system b interacting witha two-level atom a. The state |1〉a refers to an excited stateof atom a with the energy Ea and the symbol |E) denotes afree state of atom b inside the continuum with the energy E.The symbols µa and µ stand for the dipole moments betweenthe ground states |0〉a and |0〉b and the corresponding excitedstates, respectively; αL is the pumping amplitude. The con-stant J quantifies energy transfer due to the dipole-dipoleinteraction between the states |1〉a and |E); the double arrowindicates that two electrons at the atoms a and b participatein a given interaction.

equations are solved. Photoelectron ionization spectrain the long-time limit are studied in Sec. III. In Sec. IV,the profiles of long-time photoelectron ionization spec-tra are discussed and compared with those of the Fanomodel. The dynamical and Fano-like spectral zeros areinvestigated in Sec. V. Conclusions are drawn in Sec. VI.

II. QUANTUM MODEL OF THE SYSTEM AND

ITS DYNAMICS

We consider an ionization system that interacts witha neighbor two-level atom by energy transfer (for thescheme, see Fig. 1). Both atoms are under the influence

of a stationary optical field. The Hamiltonian Hion ofionization atom b with flat continuum levels interactingwith the optical field can be written in the form (h = 1is assumed, [18]):

Hion =

∫

dEE|E)(E|

+

∫

dE [µαL exp(−iELt)|E) b〈0|+H.c.] . (1)

Here, the continuum of states of atom b is formed bythe states |E) with the energies E. The dipole momentµ characterizes an optical excitation of the continuumstates; αL stands for an optical-field amplitude that os-cillates at the frequency EL. The symbol H.c. denotesthe Hermitian conjugate term. It should be noted thatthe field frequency and the energies of atomic levels arechosen in such a way that the possible excitation to thecontinuum is located far above the ionization threshold.This assumption allows to neglect the threshold effects

and it simplifies calculations of the integrals over the con-tinuum energies later.The neighbor two-level atom a with its dipole moment

µa being under the influence of the optical field is char-acterized by the Jaynes-Cummings Hamiltonian Ht−a:

Ht−a = Ea|1〉aa〈1|+ [µaαL exp(−iELt)|1〉aa〈0|+H.c.] .(2)

The state |1〉a refers to an excited state of atom a withthe energy Ea and the ground state is denoted as |0〉a.We note that the ground states of atoms a and b areassumed to have the same energy that is chosen to bezero.Energy of the interaction between the two-level atom a

and the ionization system b is given by the HamiltonianHtrans:

Htrans =

∫

dEJ |E) b〈0||0〉aa〈1|+H.c. (3)

This energy arises from the dipole-dipole interaction be-tween two electrons at the atoms a and b [17]. Contraryto the optical dipole interaction, both electrons take partin this interaction and change their states. One elec-tron returns from its excited state into the ground state,whereas the other one takes the left energy and movesfrom the ground state into its own excited state. Theconstant J quantifies this energy transfer between theatoms.We assume that the electrons at both the two-level

atom a and the ionization system b are initially in theirground states. The interaction of two electrons at theatoms a and b with the optical field as well as their mutualinteraction lead to the evolution that can be convenientlydescribed in the rotating frame. In this frame, a quantumstate |ψ〉 can be decomposed into the following generalform:

|ψ〉(t) = c00(t)|0〉a|0〉b + c10(t)|1〉a|0〉b+

∫

dEd0(E, t)|0〉a|E)

+

∫

dEd1(E, t)|1〉a|E). (4)

The coefficients of decomposition, c00, c10, d0(E), andd1(E), are time dependent.The dynamics of the composite system governed by

the Schrodinger equation with the Hamiltonian Hion +Ht−a+ Htrans can be conveniently described using differ-ential equations for the coefficients of decomposition ofthe state |ψ〉 in Eq. (4):

id

dt

[

c(t)d(E, t)

]

=

[

A B∫

dEB

†K(E)

] [

c(t)d(E, t)

]

.

(5)

Here, the symbol † means the Hermitian conjugation. InEq. (5), the vectors c and d and the matrices A, B, and

3

K take the form:

c(t) =

[

c00(t)c10(t)

]

, d(E, t) =

[

d0(E, t)d1(E, t)

]

, (6)

A =

[

0 µ∗aα

∗L

µaαL ∆Ea

]

, (7)

B =

[

µ∗α∗L 0

J∗ µ∗α∗L

]

, (8)

K(E) =

[

E − EL µ∗aα

∗L

µaαL E − EL +∆Ea

]

; (9)

∆Ea = Ea−EL. We note that the norm of state |ψ〉(t) isconserved during the evolution, which implies the equal-ity:

1∑

j=0

|cj0(t)|2 +1

∑

j=0

∫

dE |dj(E, t)|2 = 1. (10)

The system of differential equations (5) can be analyt-ically solved using the Laplace transform. TransformingEqs. (5), we arrive at the following system of algebraicequations (ε > 0):

(

ε14 −[

A B∫

dEB

†K(E)

])[

c(ε)

d(E, ε)

]

= i

[

c(0)02

]

, (11)

where the vector c(0) ≡ c(t = 0) describes the initialconditions. The symbol 1k stands for the unity matrixin k dimensions, whereas the symbol 0k means the zerovector with k elements.The coefficients d0(E, ε) and d1(E, ε) describing the

continuum can be determined from the last two equationsin (11):

d(E, ε) = [ε12 −K(E)]−1B

†c(ε). (12)

The substitution of Eqs. (12) into the first two equations(11) leaves us with two equations only for the coefficientsc00 and c10:

(

ε12 −A−B

∫

dE [ε12 −K(E)]−1

B†

)

× c(ε) = ic(0). (13)

The inverse matrix [ε12 −K(E)]−1

occurring inEq. (13) can be decomposed conveniently as follows:

[ε12 −K(E)]−1 =2

∑

k=1

Kk

1

E − ε− ξk. (14)

The energies ξ1 and ξ2 give oscillations of the two-levelatom a:

ξ1,2 = EL − ∆Ea ± δξ

2,

δξ =√

(∆Ea)2 + 4|µaαL|2. (15)

The matrices K1 and K2 introduced in Eq. (14) have theform:

Kk =(−1)k

δξ

[

Ea + ξk −µ∗aα

∗L

−µaαL EL + ξk

]

. (16)

The integration of equation (14) over the energies E ofcontinuum from −∞ to ∞ provides a useful relation thatconsiderably simplifies the system of equations (13):

∫

dE [ε12 −K(E)]−1 = −iπ12. (17)

Equation (13) then represents an eigenvalue problemfor a newly defined matrix M:

(ε12 +M) c(ε) = ic(0), (18)

M = A− iπBB†. (19)

The substitution from Eqs. (7) and (8) provides the ma-trix M:

M =

[

−iπ|µαL|2 M caαL

MaαL ∆Ea − iπ(|J |2 + |µαL|2)

]

; (20)

Ma = µa − iπµJ∗ and M ca = µ∗

a − iπµ∗J .Introducing a matrix ΛM with the eigenvalues ΛM,j

on its diagonal and the matrix of eigenvectors P (M =PΛMP

−1), the solution of Eq. (18) can be written asfollows:

c(ε) = iPUε(ε)P−1

c(0), (21)

where

[Uε]jk (ε) =δjk

ε− ΛM,j. (22)

The symbol δjk means the Kronecker δ.The inverse Laplace transform finally gives us the tem-

poral evolution of coefficients c00 and c10:

c(t) = PU(t)P−1c(0) (23)

and

Ujk(t) = δjk exp(−iΛM,jt). (24)

The substitution of the spectral solution for the coef-ficients c from Eq. (21) into Eq. (12) and the subsequentinverse Laplace transform leave us with the temporal co-efficients d0(E) and d1(E) of the continuum:

d(E, t) = i

2∑

k=1

KkB†PUk(E, t)P

−1c(0); (25)

[Uk]jl (E, t) =iδjl

E − ΛM,j − ξk

× [exp[i(ξk − E)t]− exp(−iΛM,jt)] . (26)

If we restrict our attention to the case in which bothelectrons are initially in their ground states |0〉a and |0〉b,

4

the solution written in Eq. (25) can be recast into a sim-plified form:

d(E, t) = i

2∑

k=1

Dkuξk(E, t), (27)

[uξk ]j (E, t) =i

E − ΛM,j − ξk[exp[i(ξk − E)t]

− exp(−iΛM,jt)] , j = 1, 2. (28)

The eigenvalues ΛM,j of matrix M can be analyticallyderived as follows:

ΛM,1,2 = Ea − iπ|µαL|2

∓√

E2a +MaM c

a|αL|2, (29)

where Ea = (∆Ea − iγa)/2. Moreover, the knowledge ofeigenvectors of the matrix M allows us to determine thematrices D1 and D2 introduced in Eq. (27). Their ma-trix elements give weights to the Lorentzian profiles thatconstitute the photoelectron ionization spectra. Thesematrix elements [Dk]jl are defined along the formula[Dk]jl = [KkB

†P]jl[P

−1]l1 that provides the followingexplicit relations:

Dk =1

δξD

[

Dk,11 Dk,12

Dk,21 Dk,22

]

; (30)

Dk,11 = [µE2 + JMaE ]αL[±∆Ea/2 + δξ/2]

∓ µµ∗aMaαL|αL|2E ,

Dk,12 = [µMaMcaαL|αL|2 − JMaαLE ]

× [±∆Ea/2 + δξ/2]± µµ∗aMaαL|αL|2E ,

Dk,21 = ∓µµaα2LE2 ∓ JµaMaα

2LE

+ µMaα2LE [∓∆Ea/2 + δξ/2],

Dk,22 = ∓µµaMaMcaα

2L|αL|2 ± JµaMaα

2LE

+ µMaα2LE [±∆Ea/2− δξ/2],

E = −Ea −√

E2a +MaM c

a|αL|2,D = −E2 −MaM

ca |αL|2.

The Rabi frequency δξ has been introduced in Eq. (15).The analysis of temporal behavior of the ionization

system has shown that the atom b is completely ionizedfor sufficiently long times. The electron from the atom bis transferred to the continuum and then left in a statethat is quantum correlated (entangled) with the state ofthe electron at the two-level atom a. Discussion of thisphenomenon will be presented elsewhere. The photoelec-tron ionization spectrum is composed of four Lorentziancurves as Eq. (27) indicates. Weights of these curveschange in time until they reach their long-time values.We note that, in general, these weights depend also on theinitial conditions for the atoms a and b. As we considerthe photoelectron ionization spectra obtained in station-ary optical fields, their long-time limit is of prominentinterest for us.

III. LONG-TIME PHOTOELECTRON

IONIZATION SPECTRA

The long-time photoelectron ionization spectral pro-files are derived from the long-time form of coefficientsd0(E, t) and d1(E, t) given in Eq. (25). In this limit, i.e.,for the times t obeying t≫ 1/|ImΛM,j| for j = 1, 2 (Immeans the imaginary part), the formulas for the evolu-tion matrices Uk(E, t) and vectors uξk(E, t) in Eqs. (26)and (28), respectively, considerably simplify:

[

Ultk

]

jl(E, t) = δjl [uξk ]j (E, t)

=iδjl exp[i(ξk − E)t]

E − ΛM,j − ξk; (31)

the superscript lt stands for long time. The expressionsfor the evolution matricesUlt

k in Eq. (31) show that thereexist two prominent frequencies ξ1 and ξ2 of periodic os-cillations in this long-time limit. That is why we decom-pose the long-time spectra d

lt of an ionized electron atthe atom b into parts dξ1 and d

ξ2 containing oscillationsat the frequencies ξ1 and ξ2, respectively:

dlt(E, t) = d

ξ1(E, t) + dξ2(E, t), (32)

dξj(E, t) = iKjB

†PU

ltk (E, t)P

−1c(0),

j = 1, 2. (33)

The form of the amplitude long-time photoelectron ion-ization spectra d

lt as written in Eq (32) means that theintensity photoelectron ionization spectra I ltj ≡ |dltj |2,j = 0, 1, can be decomposed into two contributions. Thefirst steady-state contributions denoted as Ist0 and Ist1 aretime independent, whereas the second ones with the com-mon magnitude Iosc harmonically oscillate at the Rabifrequency ξ1 − ξ2 = δξ:

I ltj (E, t) = Istj (E) + Ioscj (E) cos[δξt+ ϕj(E)],

(34)

ϕj(E) = arg[dξ1j+1(E)dξ2∗

j+1(E)],

Istj (E) = |dξ1j+1(E)|2 + |dξ2

j+1(E)|2, (35)

Ioscj (E) = 2|dξ1j+1(E)||dξ2

j+1(E)|, j = 0, 1. (36)

The symbol arg denotes the argument of a complex num-ber. It holds that ϕ0(E) = ϕ1(E) ± π and Iosc0 (E) =Iosc1 (E) ≡ Iosc(E). Thus the overall photoelectron ion-ization spectrum I lt(E) = I lt0 (E, t) + I lt1 (E, t) is time in-dependent and can be determined along the formula:

I lt(E) = Ist0 (E) + Ist1 (E). (37)

We note that the oscillations at the Rabi frequency δξ inthe long-time limit can be alternatively moved into theevolution of the state of atom a provided that a suit-able basis in the space of states of the ionized electron ischosen.In order to observe temporal oscillations in a pho-

toelectron ionization spectrum, the time-resolved spec-troscopy of ionized electrons is needed. The measure-ment of ionization spectra has also to be done in the

5

post-selection regime, after the measurement whose re-sult guarantees the presence of an electron at the atom aeither in the ground state or the excited state. The tem-poral resolution needed depends linearly on the strengthof the stationary pumping field [18]. If the experimen-tal temporal resolution is not sufficient, just steady stateparts Ist0 (E) and Ist1 (E) are observed.Two prominent features may characterize the photo-

electron ionization spectra in the long-time limit: thepresence of the Fano and dynamical zeros.The term Fano zero denotes a frequency in the photo-

electron ionization spectrum that cannot be excited forany strength of the optical pumping. The frequency EF

of a Fano zero can thus be revealed from the condition

I lt(EF ) = 0. (38)

According to Eq. (37), a Fano zero at the frequency EF

occurs provided that both Ist0 (EF ) = 0 and Ist1 (EF ) =0. In the investigated model, a genuine Fano zero doesnot exist, since we have not included into our model anauto-ionizing state. However, the Fano-like zeros can beobserved for the case of a weak optical pumping.The long-time photoelectron ionization spectral inten-

sity components Ist0 , Ist1 , and Iosc obey in general the in-equalities Ist0 (E) ≥ Iosc(E) and Ist1 (E) ≥ Iosc(E). Thismeans that there might occur the frequencies ED fulfill-ing one or both the following relations:

Istj (ED) = Iosc(ED), j = 0, 1. (39)

At these frequencies ED, the long-time photoelectronionization intensity spectrum I lt0 (E, tD) [or I lt1 (E, tD)]equals zero at specific time instants tD. We call these fre-quencies ED dynamical zeros, because they periodicallyoccur with the Rabi period 2π/δξ. They represent ananalogy to the usual Fano zeros in systems that involvean additional two-level atom a oscillating in a stationaryoptical field. It holds that the dynamical zeros in thespectra I lt0 and I lt1 occur, in general, at different frequen-cies ED. We will see later that the frequencies ED ofdynamical zeros in the long-time photoelectron spectraI lt0 and I lt1 coincide provided that the atom a is resonantlypumped.If there exists a Fano zero at the frequency EF , there

also occurs a dynamical zero at the same frequency ED =EF . This can easily be understood from the conditionswritten in Eqs. (38) and (39) using the nonnegativity ofintensities.

IV. DISCUSSION OF SHAPES OF LONG-TIME

PHOTOELECTRON IONIZATION SPECTRA

There exists a similarity of the long-time photoelectronionization spectra in the studied ionization system inter-acting with a neighbor atom and the well-known Fanomodel. That is why we first summarize the main featuresof the spectra in the Fano model and then we continueby comparing the spectra in both models.

(a)

(b)

FIG. 2. Long-time photoelectron ionization spectra I lt in (a)the Fano model (qa = γa = 0, qb = 100, γb = 1) and (b)the ionization system interacting with a neighbor (qa = 100,γa = 1, qb = γb = 0) in the regime of prevailing indirectionization for several values of pumping parameter Ω: Ω = 1(solid curve), Ω = 2 (solid curve with ∗), and Ω = 4 (solidcurve with ); qa = µa/(πµJ

∗), γa = π|J |2, qb = µb/(πµV∗),

γb = π|V |2, Ω =√4πΓ(Q+ i)µαL, Γ = γa + γb, Q = (γaqa +

γbqb)/Γ. Spectra are normalized such that∫

dEI lt(E) = 1;Ea = Eb = EL = 1.

A. Long-time photoelectron ionization spectra in

the Fano model

The photoelectron ionization spectra of the usual Fanomodel of the atom b with one discrete excited state |1〉bof the energy Eb have been discussed, e.g., in [1, 2]. Thestate |1〉b can be optically excited through the dipole mo-ment µb. It also interacts with the continuum of states|E) by the Coulomb configurational interaction that ischaracterized by a constant V . Thus, there occurs com-petition between the direct and indirect (through thestate |1〉b) ionizations. The relative strength of two ion-ization paths forms two distinct areas differing in shapesof the ionization spectra.We first assume the prevailing indirect optical ioniza-

tion (qb = µb/(πµV∗) ≫ 1). In this case the long-time

photoelectron ionization spectrum is peaked around thefrequency E being resonant with the pumping optical fre-quency EL for weaker pumping intensities [see Fig. 2(a)].Greater pump amplitudes αL lead to the Autler–Townessplitting [19] of this peak that results in a symmetricdouble-peaked spectral shape. The distance betweentwo peaks equals 2|µbαL|. This means that the largerthe pump amplitude |αL|, the greater the distance be-tween two neighbor peaks. Moreover, the widths of these

6

peaks are proportional to (γb + π|µαL|2)/2 [γb = π|V |2],i.e., their broadening with the increasing pump ampli-tude αL is observed. These conclusions can be drawnfrom the general form of amplitude ionization spectrumthat is composed of two Lorentzian curves. These curvesare centered at the frequencies ReΛF

M,1,2 + EL, where

ΛFM,1,2 can be derived in the following form:

ΛFM,1,2 = ∆Eb/2− iπ|µαL|2/2− iγb/2

∓[

(∆Eb + iπ|µαL|2 − iγb)2

+4(µb − iπµV ∗)(µ∗b − iπµ∗V )|αL|2

]1/2/2; (40)

∆Eb = Eb − EL. This formula considerably simplifiesin the discussed regime if we additionally assume γb ≤ 1and qb ≫ 1:

ΛFM,1,2 = ∆Eb/2− iπ|µαL|2/2− iγb/2∓ |µbαL|.

(41)

On the other hand, if the direct and indirect opticalionizations are comparable, a typical shape of the pho-toelectron ionization spectrum consists of one peak thatmoves towards the lower frequencies E with the increas-ing pump amplitude αL [see Fig. 3(a)]. This indicatesthat the Lorentzian curve centered at the frequency ΛF

M2

given in Eq. (40) is decisive for the spectral shape. Ingeneral, there occurs one Fano zero that originates in thedestructive interference of two ionization paths. The fre-quency EF of this zero is expressed as EF = Eb − γbqb.

B. Long-time photoelectron ionization spectra of

an ionization system interacting with a neighbor

Now we pay attention to photoelectron ionization spec-tra of the ionization system interacting with a two-levelneighbor atom. Here, in parallel to the direct ionization,the ionization of atom b may also occur after the energytransfer from the excited bound state |1〉a of atom a,that is not accompanied by the electron transfer. Thisindirect ionization path interferes with the path of di-rect ionization of the atom b. The presence of the sec-ond electron at the atom a, that undergoes the long-time stationary Rabi oscillations, substantially modifiesthe photoelectron ionization spectra of the Fano modeldiscussed above. As a consequence, the photoelectronionization spectra dlt0 (E, t) and d

lt1 (E, t) belonging to the

atom a in the ground and excited states, respectively,can be decomposed into two contributions oscillating atthe prominent frequencies ξ1 and ξ2 [see Eq. (15)]. Theoverall photoelectron ionization spectrum I lt defined inEq. (37) is constituted by four Lorentzian curves centeredaround the frequencies ReΛM,j+ ξk for j, k = 1, 2 (thesymbol Re denotes the real part of an argument).As we want to compare both models at comparable

conditions, we assume that the strengths of interactionsleading to auto-ionization are comparable, i.e. γa ≈ γb.

(a)

(b)

FIG. 3. Long-time photoelectron ionization spectra I lt in (a)the Fano model (qa = γa = 0, qb = γb = 1) and (b) theionization system interacting with a neighbor (qa = γa = 1,qb = γb = 0) in the regime of comparable direct and indirectionization paths for different values of pumping parameter Ω:Ω = 0.5 (solid curve), Ω = 1 (solid curve with ∗), and Ω = 2(solid curve with ); Ea = Eb = EL = 1.

The special case of γa ≪ γb appropriate for molecularcondensates with their dipole-dipole and Coulomb con-figurational interactions is discussed at the end of Sec. IV.Provided that the indirect ionization is much stronger

than the direct one (for this case, the Fano-like asym-metry parameter qa = µa/(πµJ) ≫ 1), the assumptionsof resonant pumping (Ea = EL) and the weaker dipole-dipole interaction (γa ≤ 1, qa ≫ 1) allows us to expressfour complex central frequencies in a simple form:

ΛM,1,2 + ξ1,2 = EL − iγa/2− iπ|µαL|2∓ |µaαL| ∓ |µaαL|. (42)

These formulas indicate that the long-time photoelectronionization spectrum consists of one central peak and twosymmetric side-peaks shifted by the frequency ±2|µaαL|.It can be shown that two side-peaks dominate the pho-toelectron spectrum for weak optical pumping, whereasthree distinct peaks can be observed for stronger pump-ing [see Fig. 2(b)]. The widths of the spectral peaks areproportional to γa/2+π|µαL|2, i.e., the greater the pumpamplitude αL the broader the peaks. Compared to thelong-time photoelectron ionization spectra of the Fanomodel, we additionally have the central peak and alsothe mutual distance of two side-peaks is twice that foundin the Fano model [see Eq. (42)]. For greater pump am-plitudes αL (γa ≪ |µαL|2), the widths of the spectralpeaks are twice compared to those of the Fano model.

7

FIG. 4. Long-time photoelectron ionization spectrum I lt

(solid curve), its steady-state components Ist0 (solid curve with) and Ist1 (solid curve with ∗), and magnitude Iosc of the har-monically oscillating part (solid curve with ) for the ioniza-tion system interacting with a neighbor under non-resonantpumping; qa = γa = 1, qb = γb = 0, Ω = 2, Ea = 1, EL = 0.8.

If the direct and indirect ionizations are comparable(qa ≈ 1), two peaks dominate the long-time photoelec-tron ionization spectrum, as demonstrated in Fig. 3(b).The first peak is located near the pumping frequencyEL, whereas the second one occurs around the frequencyEL − 2|µaαL|. The second peak thus moves towards thelower frequencies when increasing the pump amplitudeαL. This behavior is qualitatively similar to that foundin the Fano model [see Fig. 3(a)]. However, the frequencyshift is two times larger in the ionization system interact-ing with a neighbor. Moreover, the spectral peaks are twotimes broader for stronger pumping.

Contrary to the Fano model, the long-time photoelec-tron spectrum I lt can be decomposed into two parts I lt0and I lt1 that are conditioned by the presence of the elec-tron at the two-level neighbor atom a in the states |0〉aand |1〉a, respectively. The conditional spectra I ltj are

described by their steady-state components Istj and thecommon term Iosc that harmonically oscillates at theRabi frequency δξ. If the neighbor atom a is resonantlypumped, Ist0 = Ist1 . Otherwise, the two steady-state com-ponents differ in their profiles. This is illustrated inFig. 4. For this case, two dynamical zeros around therelative frequency (E − Ea)/γa = −0.6 appear in thespectrum. Indeed, a closer inspection of the curves inFig. 4 reveals that, for two frequencies in this area, theamplitude Iosc of the oscillating part is equal to the valueof either the steady-state part Ist0 or the steady-state partIst1 . Moreover, it should be mentioned that, the occur-rence times of the two dynamical zeros in the profiles ofspectra I lt0 and I lt1 are mutually shifted by half the Rabiperiod π/δξ.

If molecular condensates are considered, γa ≪ γb.In molecular condensates, the Coulomb configurationalinteraction constant V is typically in eV, whereas thedipole-dipole interaction (hopping) constant J is of theorder of 1 − 10 meV [17]. This implies that γa/γb ≈10−4 − 10−6. Due to the scaling properties of the modelthe spectral profiles obtained in the regime γa ≈ γb

FIG. 5. Long-time photoelectron ionization spectra I lt inthe ionization system interacting with a neighbor for differentvalues of pumping parameter Ω: Ω = 5 × 10−5 (solid curve),Ω = 1 × 10−4 (solid curve with ∗), and Ω = 5 × 10−4 (solidcurve with ); qa = 100, γa = 1 × 10−4, qb = γb = 0,Ea = Eb = EL = 1.

can also be observed in this case provided that the op-tical pumping described by parameter Ω is weaker byfour or six orders of magnitude. As an example, thelong-time photoelectron ionization spectra determinedfor γa = 1× 10−4 are shown in Fig. 5. Their profiles aresimilar to those valid for γa = 1 and drawn in Fig. 2(b).However, we should emphasize that, for γa ≪ 1, the caseof comparable dipole moments µa and µ of atoms a andb, respectively, is characterized by the condition qa ≫ 1.On the other hand, if qa ≈ 1 then µa ≪ µ. Small val-ues of dipole moments µa of the neighbor atom a can bereached, e.g., by the detuning of atom a from the reso-nance with the pumping field applying a dc electric field.

V. DYNAMICAL AND FANO-LIKE ZEROS

The frequencies of dynamical and Fano-like (for a weakoptical pumping) zeros are distinguished features of thelong-time photoelectron ionization spectra. We can evenfind them analytically in the limit of weak optical pump-ing (αL → 0) using perturbation expansions for the ma-trices Dj in Eq. (30) and the frequencies ΛM,j in Eq. (29)and ξj in Eq. (15).

A. Fano-like zeros

We first pay attention to a weak optical pumping thatis in resonance with the two-level neighbor atom a (Ea =EL). Under this condition, it is sufficient to express thematrices D1,2 in the first power of pump amplitude αL

and the frequencies in the zeroth power of αL. We thusobtain:

D1,2 =

[

− iJMa

γa

−µ+ iJMa

γa

± iJMa

γa

µa

|µa|±(

µ− iJMa

γa

)

µa

|µa|

]

αL

2, (43)

ΛM,1,2 + ξj = EL − iγa/2∓ iγa/2, j = 1, 2. (44)

8

As the rows of the matrices D1 and D2 in Eq. (43)are the same up to a multiplicative complex factor, weobtain the same equations for the frequencies of the Fano-like zeros both in the spectra dlt0 and dlt1 as well as in theirspectral component oscillating at the frequencies ξ1 andξ2. These equations together guarantee the fulfilment ofthe condition for the Fano zero given in Eq. (38). Theyhave the common form:

iJMa

γa(EF−l − EL + iγa)+µ− iJMa/γaEF−l − EL

= 0. (45)

The solution of Eq. (45) gives the frequency EF−l ofFano-like zero at the position:

EF−l − Ea

γa= −qa; (46)

qa = µa/(πµJ∗).

In the case of weak optical pumping that is out-of-resonance with the two-level atom a (Ea 6= EL), we dis-cuss possible positions of the Fano-like zeros separatelyfor the components oscillating at the frequencies ξ1 andξ2. As for the frequency ξ1, terms in the first power of αL

in the spectrum dlt0 as well as terms in the third powerof αL in the spectrum dlt1 lead to the same equation. Wenote that the restriction to the first power of αL in theexpressions for the frequencies ΛM,j and ξj is sufficientin this derivation. The solution of the resulting equationis independent of the detuning ∆Ea and coincides withthat derived for the resonant pumping in Eq. (46).On considering the frequency ξ2, the general formula

for the matrices Dj in Eq. (30) can be approximatedby terms written in the second power of αL both forthe spectra dlt0 and dlt1 . These terms form two differ-ent equations for the frequencies EF−l of Fano-like zeros.However, the solutions of both equations coincide and wehave:

EF−l − Ea

γa= −qa −

2∆Ea

γa+i∆Ea

qaγa. (47)

Equation (47) indicates that the non-resonant pumpingof atom a moves the energy EF−l of a possible Fano-likezero into the complex plane E. This means that therecannot occur any Fano-like zero in this case.The Fano zeros in the ionization system interacting

with a neighbor have been looked for numerically in theregime of stronger optical pumping. However, no Fanozero has been revealed.

B. Dynamical zeros

The frequencies ED of dynamical zeros in the limit ofweak optical pumping have been analyzed analyticallyin the same vein. The numerical analysis has then beenfound useful for arbitrarily strong pumping. The fre-quencies ED of dynamical zeros are given by the general

formula in Eq. (39). Equivalently, this formula can bereplaced by a more suitable one:

|dξ1j (E, t)| = |dξ2j (E, t)|, j = 0, 1. (48)

Similarly as in the case of Fano-like zeros, a separatediscussion of the resonant and non-resonant pumping isconvenient.On assuming a weak optical pumping that is resonant

with the atom a (Ea = EL), the Taylor expansions of thematrices D1 and D2 in Eq. (30) to the second power ofpump amplitude αL together with the Taylor expansionof frequencies ΛM,j and ξj to the first power of αL resultin the following equation for the frequencies ED:

∣

∣

∣

∣

JMa − µµ∗aMaαL/|µa|

ED − Ea + iγa − |µaαL|

+−iµγa − JMa + µµ∗

aMaαL/|µa|ED − Ea − |µaαL|

∣

∣

∣

∣

=

∣

∣

∣

∣

−JMa − µµ∗aMaαL/|µa|

ED − Ea + iγa + |µaαL|

+iµγa + JMa + µµ∗

aMaαL/|µa|ED − Ea + |µaαL|

∣

∣

∣

∣

. (49)

We note that this equation is valid for both the spectradlt0 and dlt1 , i.e., the frequencies of dynamical zeros inthese spectra coincide.Equation (49) can be recast into a polynomial of the

fifth order in the normalized frequency E, E = (E −EL)/γa:

|pa|2ImpaE5 + [−2Repa+ Imp2a]E4

+ [|pa|2 + |pa|2Impa − 2]E3

+ Imp2aE2 − E = 0; (50)

pa = 1/qa. This means that up to five dynamical zeroscan occur, in principle, at the same positions in the spec-tra I lt0 and I lt1 . One root of the polynomial in Eq. (50)equals zero. Provided that the parameter qa is real,the remaining fourth-order polynomial collapses into thethird-order polynomial:

E3 +

(

qa −1

2qa

)

E2 +qa2

= 0. (51)

The direct inspection gives one root of this polynomialat the normalized frequency E = −qa. This frequencycoincides with the frequency of Fano-like zero given inEq. (46). The remaining two roots can then be recoveredeasily. Thus, we arrive at the following four normalizedfrequencies of dynamical zeros:

ED − Ea

γa= 0,

ED − Ea

γa= −qa,

ED − Ea

γa=

1

4qa± 1

4

√

1

q2a− 8. (52)

9

(a)

(b)

(c)

FIG. 6. Normalized frequencies (ED − Ea)/γa of dynamicalzeros as they depend on pumping parameter Ω for differentvalues of parameter qa: (a) qa = 0.1, (b) qa = 1, and (c)qa = 3; γa = 1, qb = γb = 0, Ea = EL = 1.

The last two normalized frequencies written in Eq. (52)

are real provided that |qa| ≤ 1/(2√2).

In the case of arbitrarily strong optical pumping, upto five dynamical zeros can be discovered. They usu-ally occur in pairs. When studying the dependence offrequencies ED on the pumping strength Ω (for the def-inition, see the caption to Fig. 2, Ω ∝ αL), the creationand annihilation of frequency pairs has been observed(see Fig. 6). We have found three qualitatively differ-ent shapes of curves in graphs showing the dependenceof normalized frequencies (ED −Ea)/γa on the pumpingparameter Ω depending on the relative strengths of directand indirect ionization paths. They are plotted in Fig. 6.In Fig. 6, the values of the frequencies ED for Ω → 0 co-incide with those analytically described in Eq. (52). Wealso note that the graphs in Fig. 6 are symmetric withrespect to the exchange Ω by −Ω.

On considering the non-resonant pumping of atom a(Ea 6= EL), the frequencies ED of dynamical zeros, in

general, differ for the photoelectron ionization spectra I lt0and I lt1 . On assuming a weak optical pumping αL andthe spectrum I lt0 , the Taylor expansion of the matricesDj, j = 1, 2, in Eq. (30) up to the first power in αL issufficient. The condition for dynamical zeros in Eq. (48)then reveals two normalized frequencies (ED − Ea)/γa:

ED − Ea

γa= −qa,

ED − Ea

γa= −2

∆Ea

γa. (53)

On the other hand, the Taylor expansion of matricesDj, j = 1, 2, up to the second power in αL is needed whenstudying the frequencies ED in the photoelectron ion-ization spectrum I lt1 . The condition for dynamical zeroswritten in Eq. (48) can be recast into a fifth-order polyno-mial in the frequencies ED. On assuming the real param-eter qa, this polynomial reduces to the fourth-order poly-nomial in the normalized frequency E = (ED − EL)/γa:

[

4qa +1

q2aδEa

]

E4 +

[

−2 + 4q2a +2

q2aδE2

a

]

E3

−[(

3 +1

q2a

)

δEa + 4qaδE2a +

1

q2aδE3

a

]

E2

+[

2q2a − 2qaδEa

]

E +[

q2aδEa − 2qaδE2a + δE3

a

]

= 0,

(54)

where δEa = ∆Ea/γa. We note that ∆Ea 6= 0 wasassumed in the derivation of Eq. (54). There exist, ingeneral, four roots of this polynomial that can be foundnumerically. As only real roots give the frequencies ED,there might occur 0, 2, or 4 frequencies ED determiningpositions of dynamical zeros.

Typical behavior of the dynamical zeros for the non-resonant pumping of atom a considered as a function ofthe pumping parameter Ω is shown in Fig. 7 for qa =1. The comparison with the graph in Fig. 6(b) revealstwo characteristic features caused by the non-resonantpumping. First, the curves observed for the resonantpumping are split into two adjacent ones: one belongsto the normalized frequencies of dynamical zeros in thespectra I lt0 , one gives the normalized frequencies in thespectra I lt1 . Second, there occur new curves with shapesdepending qualitatively on the sign of the detuning ∆Ea.The symmetry Ω ↔ −Ω is also preserved.

We would like to emphasize a prominent role of pa-rameter γa in the determination of frequencies ED of thedynamical zeros. The parameter γa giving the strengthof ’non-optical’ auto-ionization interaction (e.g., dipole-dipole interaction) occurs in the calculations only as ascaling parameter of the frequency difference ED − Ea.This means that the obtained results are applicable, af-ter appropriate rescaling, also to the case of molecularcondensates in which the dipole-dipole interaction givesγa ≈ 10−4 − 10−6.

10

(a)

(b)

FIG. 7. Normalized frequencies (ED − Ea)/γa of dynamicalzeros as they depend on pumping parameter Ω for the non-resonant pumping of atom a: (a) EL = 0.8 and (b) EL = 1.1.Solid curves indicate the frequencies found in the spectra I lt0 ,solid curves with correspond to the spectra I lt1 ; qa = γa = 1,qb = γb = 0, Ea = 1.

VI. CONCLUSIONS

The long-time photoelectron ionization spectra of anionization system interacting with a neighbor two-level

atom have been investigated using the Laplace-transformmethod. They have been compared with the spectracharacterizing the Fano model of auto-ionization. Thespectra are typically composed of several peaks of differ-ent widths depending on the pump-field intensity. As aconsequence of the interference of two ionization paths,zeros in the long-time photoelectron ionization spectramay occur. Whereas the genuine Fano zeros cannot befound in this model, the Fano-like zeros occurring fora weak optical pumping can be observed. The long-timephotoelectron ionization spectra conditioned by the pres-ence of the neighbor two-level atom in a given state ex-hibit the permanent Rabi oscillations. Dynamical spec-tral zeros observed once in the Rabi period have been dis-covered in these conditional spectra. The frequencies ofdynamical zeros depend on the strength of optical pump-ing as well as on the projected state of the two-level atom.The numbers as well as the values of frequencies of thedynamical zeros have been analyzed in detail. Molecularcondensates have been shown to be suitable candidatesfor the experimental verification of the predicted effect.

ACKNOWLEDGMENTS

Support by the projects 1M06002, COST OC09026, and Operational Program Research and Devel-opment for Innovations - European Social Fund (projectCZ.1.05/2.1.00/03.0058) of the Ministry of Education ofthe Czech Republic as well as the project IAA100100713of GA AV CR is acknowledged.

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